If ${}^nC_{r-1}=330$,${}^nC_r=462$,and ${}^nC_{r+1}=462$,then $r$ is equal to

The value of $\frac{{}^{10}C_{r}}{{}^{11}C_{r}}$,when both the numerator and denominator are at their greatest values,is

$A$ committee of $5$ is to be formed out of $6$ men and $4$ ladies. The number of ways this can be done,when at most $2$ ladies are included,is

$A$ person is permitted to select at least one and at most $n$ coins from a collection of $(2n + 1)$ distinct coins. If the total number of ways in which he can select coins is $255$,then $n$ equals

At an election,a voter may vote for any number of candidates not exceeding the number to be elected. If $4$ candidates are to be elected out of the $12$ contested in the election and a voter votes for at least one candidate,then the number of ways in which a voter can vote is:

If $\binom{18}{15} + 2\binom{18}{16} + \binom{17}{16} + 1 = \binom{n}{3}$,then find the value of $n$.